We must stress that this energy flow is, by construction, a property of the source, not of geometry (even for \(\chi =0\) ). The crucial point here is that the mass of the BH grows due to the redistribution of the energy density carried by the matter surrounding the BH. Whether such an energy flux is realized is ultimately determined by the properties of dark matter. No dark matter candidate is known to support this specific energy flow (especially if it is a pure heat flow). For example, in [1]} we argue that if all dark matter consists of PBHs, then the surrounding matter making up \(\rho _{M}\)  is not described by a smooth stress-energy tensor and is not capable of causing such an energy flux. The issues with the scenario where all dark matter consists of PBHs were not addressed in the comment [2]}.
